The Count and Sum options in the Summary Statistics menu are available for conditional counts and sums of the data. Consider the situation where you have 2,500 observations and need to know how many values are less than or equal to 10.0. Perform this calculation by clicking on Count , followed by clicking on the IF < = box, and then type the target value in the right hand box 10.0. The conditional count will appear with the other statistics.

The summary and test statistics are calculated using Excel functions so you can highlight the cells containing the statistics and drag (or copy/paste) the formulas to other locations. For example, if you need the statistics for 20 variables that are in columns B though U. Calculate the Summary Statistics for the data in column B and then drag the formulas to cells C through U. A sample output is provided in Step 1 of DemoSimetar-Data.

Reverse Order Function.
The order of the numbers in any array can be reversed using the array function =REVERSE( ). This function is useful for reversing a time series vector from newest value first to oldest value first. The function is used as follows: first highlight a blank column (or row) that matches the size of the original data, second type the command.

=REVERSE (Vector to Reverse)

and press the keys Control Shift Enter. This function can also be accessed by using the matrix operations icon Data manipulation and matrix operation functions and selecting the Reverse a Column or Row of Values option. See Step 18 in DemoSimetar-Mat for an example.

where: Array to Sort is an array or matrix of values to sort, Index Number is an integer indicating the column or row to be used for the sort, Ascending Order is an optional ‘TRUE or 1’ for an ascending order sort and ‘FALSE or 0’ for a descending order sort, and Sort Rows is an optional value to indicating orientation of the data: ‘TRUE or 1’ indicates columns will be sorted and ‘FALSE or 0’ indicates rows will be sorted.

Concatenate Function.
The table lookup functions in Excel (VLOOKUP and HLOOKUP) are very useful in model development, but they require the data to be adjacent in the worksheet. This restriction often results in duplication of tables and data throughout the worksheet. To minimize this duplication, use the concatenate function. The =CONCAT() function allows you to specify a virtual table for use in table lookup functions. For example, if column A1:A20 contains the numbers 1-20 and column E1:E20 contains the searchable data the =CONCAT( ) can be used in VLOOKUP to locate in column 2 that is associated with the value 10 as follows:

=VLOOKUP (10, CONCAT (A1:A20, E1:E20), 2)

An example of =CONCAT( ) to construct a new matrix and in a VLOOKUP is included as Step 19 in DemoSimetar-Mat.

Delete Text in a Cell.
Delete Numbers in a Cell. When a cell has both numbers and text and you want the text, use the =DELNUM( ) function to extract the text. If cell A1 contains the string “1013 Sycamore Street” and you want the text in cell A2, then in A2 type:

=DELNUM(A1)

An example is provided in Step 14 of DemoSimetar-Data.

Box-Cox Transformation.
The =BOXCOX() function can be used to transform the data for a skewed distribution to make it approximately normally distributed. The function uses a user specified exponent to transform the data. The =BOXCOXEXP() function is provided to estimate an appropriate exponent. The format for the Box Cox transformation functions are:

=BOXCOX( Data Location, Power Value, [Shift to Plus])

where: Data Location refers to the location of the data series to be transformed, Power Value is the exponent for the transformation, and Shift to Plus is an optional term if the data are to be shifted to positive values enter ‘TRUE or 1’, otherwise enter ‘FALSE or 0’.

The =BOXCOX( ) function is an array function so highlight the appropriate number of cells and type the function on press Control Shift Enter. See Step 15 of DemoSimetar-Data for an example. Once a model has been estimated using a Box-Cox transformation, the =UNBOXCOX() function can be used to transform the forecast values back to original data. The reverse transformer function is:

=UNBOXCOX(Data Value, Power Value, Original Data Range, [Shift to Plus])

where: Data Value is a data point or data series that needs to be transformed back to the original distribution, Power Value is the exponent for the transformation, Original Data Range is the location for the original data that was transformed, and Shift to Plus is an optional term if the data are to be shifted to positive values enter ‘TRUE or 1’, otherwise enter ‘FALSE or 0’.

The maximum likelihood estimation of the Box-Cox transformation exponent function can be accomplished using the following function:

where: Data Location refers to the location of the data series to be transformed, Shift to Plus is an optional term if the data are to be shifted to positive values, Lower Bound is an optional minimum for the search routine, -2 is the default, Upper Bound is an optional maximum for the search routine, +2 is the default, and Max Iterations is an optional parameter for the search routine.

See Step 15 of DemoSimetar-Data for an example of all three BoxCox functions.

Center Matrix of a Specified Dimension
Cofactor of a Square Matrix
Column Vector to a Diagonal Matrix
Column Vector to a Matrix
Column Vector to a Toeplitz Matrix
Determinant of a Square Matrix
Eigenvalues of a Square Matrix
Eigenvectors of a Square, Symmetric Matrix
Equicorrelation Matrix of a Specified Dimension
Exponential Power of a Matrix
Factor a Square, Symmetric Matrix
Generalized Inverse of a Matrix
Inner Product of Two Matrices
Invert a Nonsingular Square Matrix
Kronecker Multiply Two Matrices
Mahalanabis Distance of Two Data Matrices

Matrix of 1s
Matrix to a Vector
Multiply Two Matrices
Norm of a Matrix
Orthoganalize a Matrix
Rank of a Matrix
Reduced Row Echelon Form of a Matrix
Reverse a Column or Row of Values
Row Echelon Form of a Matrix
Sequence of Numbers
Sort a Matrix by a Specified Column
Sweep a Square Matrix on a Diagonal Element
Trace of a Square Matrix
Transpose a Matrix
Wishart Matrix of Random Variables

where: Data series is the location of the vector, No. of rows is the number of rows for the matrix, No. of columns is the number of columns for the matrix, Fill order is ‘1’ to fill by row and ‘0’ to fill by column. The =BLOCKIT( ) function is an array function so you must press Control Shift Enter after typing the right hand parenthesis. An example of converting a vector to a matrix is provided in Step 8 of DemoSimetar-Mat. This function is one which benefits from using Excel’s equation editor, as demonstrated in Figure 4. If you are not familiar with Excel’s equation editor, read Section 17.

Figure 4. Example of Using Excel’s Equation Editor to Program the BLOCKIT Function.

highlight a square matrix on the worksheet that is the same size as the matrix to be factored, type =MSQRT (Location of Square Matrix), an example might be = MSQRT (C24:G29), and press the Control Shift Enter keys. This three key step is required because =MSQRT( ) is an array function.

Generalized Inverse of a Rectangular Matrix.
The Generalized Inverse of a Matrix function in the Matrix Operations dialog box uses Simetar’s function. Select this option and then specify the input matrix (highlight only the numbers) and the output range for the upper left hand value, then click OK. The inverse of the input matrix will appear in the worksheet without row/column names. Copy and paste in the names if needed. An example of inverting a square matrix is provided in Step 22 of DemoSimetar-Mat.

Sweep a Square Matrix on a Diagonal Element.
The Sweep a Square Matrix on a Diagonal Element option in the Matrix Operations dialog box performs this matrix operation. Select this option and then specify the input matrix (highlight only the numbers), the reference diagonal element row number (an integer), and the output range for the upper left hand value, then click OK. See Step 23 in workbook DemoSimetar-Mat for an example.

Transpose a Matrix (Excel).
A matrix can be transposed by selecting the Transpose a Matrix option in the Matrix Operations dialog box, specifying the matrix to transpose and the upper-left hand cell to anchor the output matrix. This procedure simplifies Excel’s TRANSPOSE function by eliminating the need to block the area for the transposed matrix. See Step 6 of DemoSimetar-Mat for an example.

Multiply Two Matrices (Excel).
Excel’s matrix multiplication, MMULT, function is made easier by selecting the Multiply Two Matrices option in the Matrix Operations dialog box. The function is demonstrated in Step 10 of DemoSimetar-Mat.

Kronecker Multiply Two Matrices.
The Kronecker product of two matrices is accomplished by selecting this option in the Matrix Operations dialog box. The function is demonstrated in Step 12 in the workbook DemoSimetar-Mat.

Find the Determinant of a Square Matrix.
The determinant of a square matrix can be calculated by selecting the Determinant of a Square Matrix option in the Matrix Operations dialog box. The Excel function =MDETERM (square matrix) is used for this calculation. An example is provided in Step 5 of DemoSimetar-Mat.

Find the Eigen Values of a Square Matrix.
The Eigen values for a square matrix can be calculated by selecting the Eigenvalues of a Square Matrix option in the Matrix Operations dialog box. An example is provided in Step 24 of DemoSimetar-Mat.

Find a Cofactor of a Square Matrix.
The cofactor of a square matrix can be calculated by selecting the Cofactor of a Square Matrix option in the Matrix Operations dialog box. An example is provided in Step 25 of DemoSimetar-Mat.

Find the Rank of a Matrix.
The rank of a matrix can be calculated to determine if the matrix is singular. If an NxN matrix has an N rank if it is nonsingular. The rank of a square matrix can be calculated using the =MRANK( ) function or the Matrix Operations dialog box Rank of a Matrix option. An example of this function is provided in Step 16 of the DemoSimetar-Mat workbook.

Find the Norm of a Matrix.
The norm (the square root of the inner product of the matrix) can be calculated using the find the Norm of a Matrix option in the Matrix Operations dialog box. This option uses the =MNORM( ) function and is demonstrated in Step 9 of the DemoSimetar-Mat workbook.

Find the Inner Product of Two Matrices.
The inner product of two matrices can be calculated directly using this option in the Matrix Operations dialog box. If two matrices are of the same dimension, MxN, the inner product scalar is the sum of the products of the corresponding elements. The scalar can be calculated directly using the function =MIP(matrix 1, matrix 2) and is demonstrated in Step 11 of DemoSimetar-Mat.

Create a Sequence of Numbers.
A sequence of numbers in an array can be created using the =SEQ( ) function. The =SEQ( ) returns a column of numbers that follow any sequence you specify. The function is programmed as:

=SEQ(No. of Values, Starting Value, Interval or Increment)

where: No. of Values is the number of cells to be highlighted, Starting Value is the first value in the sequence, and Interval or Increment is the interval between each value.

For example the sequence of number for 10, 20, 30, …, 200 is generated by programming the function as =SEQ(20,10,10) and a sequence of 2, 4, 6, …, 20 is generated by programming the function as =SEQ(10, 2, 2). See Step 30 in DemoSimetar-Mat for an example.

Create a Centering Matrix. The =MCENTER() array function that creates an NxN centering matrix when N is specified as the dimension. See Step 28 in DemoSimetar-Mat for an example.

Create an Equicorrelation Matrix. The =MEQCORR() array function generates an NxN equicorrelation matrix using any specified correlation coefficient. The =MEQCORR( ) function is an array function so you must highlight the cells for the square equicorrelation matrix and end the function by hitting Control Shift Enter. The function is programmed as =MEQCORR( Rho). Where Rho is the correlation coefficient. See Step 27 in DemoSimetar-Mat for an example of a 4X4 equicorrelation matrix using 0.345 as the correlation coefficient.

Create a Toeplitz Matrix. The =MTOEP() array function creates a square symmetric Toeplitz matrix given a column or row of data. To create a Toeplitz matrix of an array in A1:A4, highlight a 4X4 array and type the function as =MTOEP(A1:A4). Be sure to press Control Shift Enter as this is an array function. See Step 26 in DemoSimetar-Mat for an example.